Microlocally correct photoacoustic reconstructions from spatially reduced data
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We investigate the inverse source problem for the wave equation arising in photoacoustic and thermoacoustic tomography. If one assumes that the speed of sound is a known constant, then there are many previously developed inversion formulas for the solution of this problem. These formulas require that the data is measured on a closed surface completely surrounding the object or on certain unbounded surfaces. However, in many practical applications, data can only be measured on a finite open surface. In this talk, we will present a non-iterative approach to this problem that yields an approximate solution under certain geometric restrictions on finite open surfaces. This solution coincides with the true solution microlocally--that is, the error is a smooth function. In practical applications, such reconstructions result in qualitatively correct images. For the case of a circular acquisition surface, our method can be implemented with explicit, asymptotically fast methods. We demonstrate the work of these algorithms in a series of numerical simulations.